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RAMADHANI UTAMI, NANDA NINGTYAS, I. NYOMAN WIDANA, and NI MADE ASIH. "PERBANDINGAN SOLUSI SISTEM PERSAMAAN NONLINEAR MENGGUNAKAN METODE NEWTON-RAPHSON DAN METODE JACOBIAN." E-Jurnal Matematika 2, no. 2 (May 31, 2013): 11. http://dx.doi.org/10.24843/mtk.2013.v02.i02.p032.
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System of nonlinear equations is a collection of some nonlinear equations. The Newton-Raphson method and Jacobian method are methods used for solving systems of nonlinear equations. The Newton-Raphson methods uses first and second derivatives and indeed does perform better than the steepest descent method if the initial point is close to the minimizer. Jacobian method is a method of resolving equations through iteration process using simultaneous equations. If the Newton-Raphson methods and Jacobian methods are compared with the exact value, the Jacobian method is the closest to exact value but has more iterations. In this study the Newton-Raphson method gets the results faster than the Jacobian method (Newton-Raphson iteration method is 5 and 58 in the Jacobian iteration method). In this case, the Jacobian method gets results closer to the exact value.
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Kornerup, Peter, and Jean-Michel Muller. "Choosing starting values for certain Newton–Raphson iterations." Theoretical Computer Science 351, no. 1 (February 2006): 101–10. http://dx.doi.org/10.1016/j.tcs.2005.09.056.
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Shayya, W. H., R. H. Mohtar, and M. S. Baasiri. "A Computer Model for the Hydraulic Analysis of Open Channel Cross Sections." Journal of Agricultural and Marine Sciences [JAMS] 1 (January 1, 1996): 57. http://dx.doi.org/10.24200/jams.vol1iss0pp57-64.
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Irrigation and hydraulic engineers are often faced with the difficulty of tedious trial solutions of the Manning equation to determine the various geometric elements of open channels. This paper addresses the development of a computer model for the design of the most commonly used channel-sections. The developed model is intended as an educational tool. It may be applied to the hydraulic design of trapezoidal , rectangular, triangular, parabolic, round-concered rectangular, and circular cross sections. Two procedures were utilized for the solution of the encountered implicit equations; the Newton-Raphson and the Regula-Falsi methods. In order to initiate the solution process , these methods require one and two initial guesses, respectively. Tge result revealed that the Regula-Flasi method required more iterations to coverage to the solution compared to the Newton-Raphson method, irrespective of the nearness of the initial guess to the actual solution. The average number of iterations for the Regula-Falsi method was approximately three times that of the Newton-Raphson method.
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Souza, Luiz Antonio Farani de, Emerson Vitor Castelani, and Wesley Vagner Inês Shirabayashi. "Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems." Semina: Ciências Exatas e Tecnológicas 42, no. 1 (June 2, 2021): 63. http://dx.doi.org/10.5433/1679-0375.2021v42n1p63.
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In this paper we adapt the Newton-Raphson and Potra-Pták algorithms by combining them with the modified Newton-Raphson method by inserting a condition. Problems of systems of sparse nonlinear equations are solved the algorithms implemented in Matlab® environment. In addition, the methods are adapted and applied to space trusses problems with geometric nonlinear behavior. Structures are discretized by the Finite Element Positional Method, and nonlinear responses are obtained in an incremental and iterative process using the Linear Arc-Length path-following technique. For the studied problems, the proposed algorithms had good computational performance reaching the solution with shorter processing time and fewer iterations until convergence to a given tolerance, when compared to the standard algorithms of the Newton-Raphson and Potra-Pták methods.
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Mandailina, Vera, Syaharuddin Syaharuddin, Dewi Pramita, Malik Ibrahim, and Habib Ratu Perwira Negara. "Wilkinson Polynomials: Accuracy Analysis Based on Numerical Methods of the Taylor Series Derivative." Desimal: Jurnal Matematika 3, no. 2 (May 28, 2020): 155–60. http://dx.doi.org/10.24042/djm.v3i2.6134.
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Some of the numeric methods for solutions of non-linear equations are taken from a derivative of the Taylor series, one of which is the Newton-Raphson method. However, this is not the only method for solving cases of non-linear equations. The purpose of the study is to compare the accuracy of several derivative methods of the Taylor series of both single order and two-order derivatives, namely Newton-Raphson method, Halley method, Olver method, Euler method, Chebyshev method, and Newton Midpoint Halley method. This research includes qualitative comparison types, where the simulation results of each method are described based on the comparison results. These six methods are simulated with the Wilkinson equation which is a 20-degree polynomial. The accuracy parameters used are the number of iterations, the roots of the equation, the function value f (x), and the error. Results showed that the Newton Midpoint Halley method was the most accurate method. This result is derived from the test starting point value of 0.5 to the equation root x = 1, completed in 3 iterations with a maximum error of 0.0001. The computational design and simulation of this iterative method which is a derivative of the two-order Taylor series is rarely found in college studies as it still rests on the Newton-Raphson method, so the results of this study can be recommended in future learning.
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Sabziev, Elkhan Nariman. "Calculation of Flight Data Fusion Coefficients Using Newton–Raphson Iterations." Modeling, Control and Information Technologies, no. 4 (October 23, 2020): 100–103. http://dx.doi.org/10.31713/mcit.2020.20.
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The problem of plotting the flight path of an aircraft based on flight data containing numerous measurement errors is investigated. A theoretical (continuous) model of the flight data fusion problem is proposed in the form of a boundary value problem for a system of differential equations with unknown coefficients. The application of the Newton–Raphson iteration method for calculating the sought-for coefficients is described.
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Tong, Ming Yu, Di Jian Xu, Jin Liang Shi, and Yan Shi. "Application of the Newton-Raphson Method in a Voice Localization System." Applied Mechanics and Materials 513-517 (February 2014): 4435–38. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.4435.
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In a voice localization system, the nonlinear equation of voice sources coordinates were established according the accepted information, so the algorithm for solving nonlinear equations is the key problem to the voice source localization syetem. In this paper, the Newton - Raphson method (N-R) is applied to solve the nonlinear equations, by setting the initial solution of equations,then calculating the unbalance vector and Jacobian matrix (J) so as to attain corrected vetor, after modification and iteration the initial solution,until meet the precision numerical solution. Test results show that, application of N-R method in voice localization system have advantange of less number of iterations, saving chip resources and high precision, can meet the precision requirements of voice localization system.
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Raposo-Pulido, V., and J. Peláez. "An efficient code to solve the Kepler equation." Astronomy & Astrophysics 619 (November 2018): A129. http://dx.doi.org/10.1051/0004-6361/201833563.
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Context. This paper introduces a new approach for solving the Kepler equation for hyperbolic orbits. We provide here the Hyperbolic Kepler Equation–Space Dynamics Group (HKE–SDG), a code to solve the equation. Methods. Instead of looking for new algorithms, in this paper we have tried to substantially improve well-known classic schemes based on the excellent properties of the Newton–Raphson iterative methods. The key point is the seed from which the iteration of the Newton–Raphson methods begin. If this initial seed is close to the solution sought, the Newton–Raphson methods exhibit an excellent behavior. For each one of the resulting intervals of the discretized domain of the hyperbolic anomaly a fifth degree interpolating polynomial is introduced, with the exception of the last one where an asymptotic expansion is defined. This way the accuracy of initial seed is optimized. The polynomials have six coefficients which are obtained by imposing six conditions at both ends of the corresponding interval: the polynomial and the real function to be approximated have equal values at each of the two ends of the interval and identical relations are imposed for the two first derivatives. A different approach is used in the singular corner of the Kepler equation – |M| < 0.15 and 1 < e < 1.25 – where an asymptotic expansion is developed. Results. In all simulations carried out to check the algorithm, the seed generated leads to reach machine error accuracy with a maximum of three iterations (∼99.8% of cases with one or two iterations) when using different Newton–Raphson methods in double and quadruple precision. The final algorithm is very reliable and slightly faster in double precision (∼0.3 s). The numerical results confirm the use of only one asymptotic expansion in the whole domain of the singular corner as well as the reliability and stability of the HKE–SDG. In double and quadruple precision it provides the most precise solution compared with other methods.
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Yen, Jerome, Bangren Chen, KangZhang Wu, and Joseph Yen. "Fast generation of implied volatility surface: Optimize the traditional numerical analysis and machine learning." International Journal of Financial Engineering 08, no. 02 (June 2021): 2150037. http://dx.doi.org/10.1142/s2424786321500377.
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Machine learning has been used in financial markets in supporting many tasks, such as, asset movement forecasting and trading signal generation. Monte Carlo simulation and traditional numerical methods like Newton–Raphson have also been widely applied in financial markets, such as calculation for implied volatility (IV) and pricing of financial products. Is it possible to combine such approaches to more efficiently calculate the IVs to support the generation of IV surface, term structure, and smile? In this paper, we propose a framework that combines the traditional approaches and modern machine learning to support such calculation. In addition, we also propose an adaptive Newton–Raphson to reduce the number of iterations and the possibility of falling into local minimal over the traditional Newton–Raphson. Combining the superiorities of modern machine learning and adaptive Newton–Raphson, an improvement on computation efficiency over pure traditional numerical approaches was achieved. In addition, we also take into consideration of migrating such computation to hardware accelerators such as Graphics cards (GPU) and Field Programmable Gate Arrays (FPGA), to further speed up the computation. Therefore, polynomial regression has also been tested to generate the initial guess of IVs to pave the road of such migration.
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Zotos, Euaggelos E., Satyendra Kumar Satya, Rajiv Aggarwal, and Sanam Suraj. "Basins of Convergence in the Circular Sitnikov Four-Body Problem with Nonspherical Primaries." International Journal of Bifurcation and Chaos 28, no. 05 (May 2018): 1830016. http://dx.doi.org/10.1142/s0218127418300161.
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The Newton–Raphson basins of convergence, related to the equilibrium points, in the Sitnikov four-body problem with nonspherical primaries are numerically investigated. We monitor the parametric evolution of the positions of the roots, as a function of the oblateness coefficient. The classical Newton–Raphson optimal method is used for revealing the basins of convergence, by classifying dense grids of initial conditions in several types of two-dimensional planes. We perform a systematic and thorough analysis in an attempt to understand how the oblateness coefficient affects the geometry as well as the basin entropy of the convergence regions. The convergence areas are related with the required number of iterations and also with the corresponding probability distributions.
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Souza, Luiz Antonio Farani de, Emerson Vitor Castelani, Wesley Vagner Inês Shirabayashi, Angelo Aliano Filho, and Roberto Dalledone Machado. "Trusses Nonlinear Problems Solution with Numerical Methods of Cubic Convergence Order." Tema (São Carlos) 19, no. 1 (May 5, 2018): 161. http://dx.doi.org/10.5540/tema.2018.019.01.161.
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A large part of the numerical procedures for obtaining the equilibrium path or load-displacement curve of structural problems with nonlinear behavior is based on the Newton-Raphson iterative scheme, to which is coupled the path-following methods. This paper presents new algorithms based on Potra-Pták, Chebyshev and super-Halley methods combined with the Linear Arc-Length path-following method. The main motivation for using these methods is the cubic order convergence. To elucidate the potential of our approach, we present an analysis of space and plane trusses problems with geometric nonlinearity found in the literature. In this direction, we will make use of the Positional Finite Element Method, which considers the nodal coordinates as variables of the nonlinear system instead of displacements. The numerical results of the simulations show the capacity of the computational algorithm developed to obtain the equilibrium path with force and displacement limits points. The implemented iterative methods exhibit better efficiency as the number of time steps and necessary accumulated iterations until convergence and processing time, in comparison with classic methods of Newton-Raphson and Modified Newton-Raphson.
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Hindmarsh, Richard C. A., and Antony J. Payne. "Time-step limits for stable solutions of the ice-sheet equation." Annals of Glaciology 23 (1996): 74–85. http://dx.doi.org/10.1017/s0260305500013288.
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Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.
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Hindmarsh, Richard C. A., and Antony J. Payne. "Time-step limits for stable solutions of the ice-sheet equation." Annals of Glaciology 23 (1996): 74–85. http://dx.doi.org/10.3189/s0260305500013288.
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Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.
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Blokhina, Nina S. "Physical Nonlinearity and Anisotropic Features of Materials in Structure Analysis." Applied Mechanics and Materials 405-408 (September 2013): 2686–89. http://dx.doi.org/10.4028/www.scientific.net/amm.405-408.2686.
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A justified assessment of the work of constructions and their elements should take into account physical and mechanical characteristics of construction materials. Although taking physical nonlinearity and anisotropic features into consideration will make the design more complicated, it is strongly needed to develop tools and methods that cover all the specific features of the material more precisely. Mathematical methods of construction design that take physical nonlinearity into account are quite well investigated. They are: Newton Raphson method; modified Newton Raphson method which differs from the original version by the fact that the stiffness matrix has to be calculated only once at the first iteration and remains the same for several further iterations and steps of load; the method of growing stiffness which is quite efficient in dealing with problems of physical nonlinearity, and so on. But certain difficulties appear as we consider these problems from the point of view of physics: the majority of strength and plastic-yield criteria are not implemented in design process yet. That is caused by complicated mathematics and a need for more experiment in order to determine the constants in use.
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Datangeji, Richard Umbu, Ali Warsito, Hadi Imam Sutaji, and Laura A. S. Lapono. "KAJIAN DISTRIBUSI INTENSITAS CAHAYA PADA FENOMENA DIFRAKSI CELAH TUNGGAL DENGAN METODE BAGI DUA DAN METODE NEWTON RAPHSON." Jurnal Fisika : Fisika Sains dan Aplikasinya 4, no. 2 (December 16, 2019): 56–69. http://dx.doi.org/10.35508/fisa.v4i2.976.
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Abstrak
Telah dilakukan penelitian tentang distribusi intensitas cahaya pada fenomena difraksi celah tunggal dengan tujuan menerapkan metode Bagi Dua dan metode Newton Raphson untuk memperoleh solusi jarak antara dua titik intensitas dalam fenomena difraksi celah tunggal, menetukan jarak antara dua intensitas pada pita terang, memperoleh grafik distribusi intensitas cahaya terhadap jarak pada kasus difraksi cahaya Franhoufer celah tunggal, serta membandingkan kekonvergenan metode Bagi Dua dan metode Newton Raphson. Solusi jarak antara dua intensitas pada pita terang pada kasus difraksi cahaya Franhoufer celah tunggal diperoleh dengan mencari akar-akar persamaan intensitas cahayanya. Hasil penelitian menunjukan jarak yang semakin besar ketika intensitasnya makin kecil. Ada tiga puncak intensitas, yang pertama puncak untuk intensitas maksimum pada terang pusat yang berada pada jarak 0 cm dan dua puncak untuk terang pertama setelah terang pusat yang mana intensitasnya tinggal 0.05I0 dan berada pada jarak 0.154875 cm sebelah kiri dan sebelah kanan dari intensitas maksimum. Grafik antara jarak dengan perbandingan intensitas terhadap terang maksimum berbentuk sinusoidal, terdapat tiga puncak intensitas. Puncak pertama menunjukan intensitas maksimum yang terdapat pada pita terang pusat dan dua puncak dengan intensitas 0.05I0 yang berada pita terang pertama. Pada kasus ini diperoleh hasil bahwa metode Newton Raphson lebih cepat konvergen dari metode Bagi Dua karena hanya memerlukan 4 iterasi untuk memperoleh solusi, sedangkan metode Bagi Dua membutuhkan 20 iterasi. Metode Newton Raphson juga memiliki nilai error pendekatan lebih kecil dari metode Bagi Dua yaitu 6.43929 x 10-13 sampai 7.52642 x 10-7 sedangkan metode Bagi Dua 1.90735 x 10-6.
Abstract
Research on the distribution of light intensity in the phenomenon of single slit diffraction has been carried out with the aim of applying the Bisection method and the Newton Raphson method to obtain a solution between two points in a single slit diffraction phenomenon, determining the distance between two point of intensity in the bright band, obtaining a graph of the light intensity distribution to distance in the case of Franhoufer single slit light diffraction, and comparing the speed of convergence of the Bisection method and the Newton Raphson method. The solution of the distance between two intensities in the bright band in the case of Franhoufer light diffraction in a single slit obtained by looking for the roots of the light intensity equation. The results of the study show that the greater the distance when then intensity gets smaller. There are three peak intensities, the first peak for the highest intensity in the central bright band which is located at a distance of 0 cm and two peaks in the first bright with the intensity is 0.05I0 and is 0.154875 cm left and right of the maximum intensity. The graph between the distance and intensity ratio is sinusoidal, which is three peak intensities. The first peak shows the highest intensity in the central bright band and the two peaks with the intensity of 0.05I0 which is the first bright band. In this case the results of the Newton Raphson method are converged faster than the method of Bisection because it only requires 4 iterations to obtain a solution, while the Bisection method requires 20 iterations. The Newton Raphson method also has a smaller error value than the Bisection method, which is 6.43929 x 10-13 to 7.52642 x 10-6 when the Bisection method is 1.90735 x 10-6.
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Moroz, Leonid, Volodymyr Samotyy, Cezary J. Walczyk, and Jan L. Cieśliński. "Fast Calculation of Cube and Inverse Cube Roots Using a Magic Constant and Its Implementation on Microcontrollers." Energies 14, no. 4 (February 18, 2021): 1058. http://dx.doi.org/10.3390/en14041058.
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We develop a bit manipulation technique for single precision floating point numbers which leads to new algorithms for fast computation of the cube root and inverse cube root. It uses the modified iterative Newton–Raphson method (the first order of convergence) and Householder method (the second order of convergence) to increase the accuracy of the results. The proposed algorithms demonstrate high efficiency and reduce error several times in the first iteration in comparison with known algorithms. After two iterations 22.84 correct bits were obtained for single precision. Experimental tests showed that our novel algorithm is faster and more accurate than library functions for microcontrollers.
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Zotos, Euaggelos E. "Basins of Convergence of Equilibrium Points in the Generalized Hill Problem." International Journal of Bifurcation and Chaos 27, no. 12 (November 2017): 1730043. http://dx.doi.org/10.1142/s0218127417300439.
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The Newton–Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter [Formula: see text] varies. The multivariate Newton–Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.
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Novák, Josef P., Vlastimil Růžička, Anatol Malijevský, Jaroslav Matouš, and Jan Linek. "The DAN method for calculation of vapour-liquid equilibria using an equation of state; Flash calculation." Collection of Czechoslovak Chemical Communications 50, no. 1 (1985): 23–32. http://dx.doi.org/10.1135/cccc19850023.
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A modification of the computational technique for flash calculations using an equation of state has been developed. The procedure consists in the double application of the Newton-Raphson method (DAN) to the set of equilibrium conditions. The algorithm is designed to minimize the number of iterations. It is, therefore, especially useful in successive calculations, where a family of solutions at slightly changing conditions is desired.
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Hosseini, Majid, Hassan Chizari, Tim Poston, Mazleena Bt Salleh, and Abdul Hanan Abdullah. "Efficient Underwater RSS Value to Distance Inversion Using the Lambert Function." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/175275.
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There are many applications for using wireless sensor networks (WSN) in ocean science; however, identifying the exact location of a sensor by itself (localization) is still a challenging problem, where global positioning system (GPS) devices are not applicable underwater. Precise distance measurement between two sensors is a tool of localization and received signal strength (RSS), reflecting transmission loss (TL) phenomena, is widely used in terrestrial WSNs for that matter. Underwater acoustic sensor networks have not been used (UASN), due to the complexity of the TL function. In this paper, we addressed these problems by expressing underwater TL via the Lambert W function, for accurate distance inversion by the Halley method, and compared this to Newton-Raphson inversion. Mathematical proof, MATLAB simulation, and real device implementation demonstrate the accuracy and efficiency of the proposed equation in distance calculation, with fewer iterations, computation stability for short and long distances, and remarkably short processing time. Then, the sensitivities of LambertWfunction and Newton-Raphson inversion to alteration in TL were examined. The simulation results showed that LambertWfunction is more stable to errors than Newton-Raphson inversion. Finally, with a likelihood method, it was shown that RSS is a practical tool for distance measurement in UASN.
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GAL, E., M. ZELKHA, and R. LEVY. "A SIMPLE CO-ROTATIONAL GEOMETRICALLY NON LINEAR MEMBRANE FINITE ELEMENT WRINKLING ANALYSIS." International Journal of Structural Stability and Dynamics 11, no. 01 (February 2011): 181–95. http://dx.doi.org/10.1142/s0219455411004038.
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Thin pre-tensioned membranes are often used in civil architecture as well as in marine and space technologies. Lacking in-plane compression stiffness, membranes will wrinkle at certain states of stress. This paper presents a geometrically nonlinear analysis of membranes in the presence of wrinkling using a unique incremental formulation that accounts for equilibrium in the deformed state and the current wrinkling state by iterations. The membranes with wrinkling are simulated by a geometrically nonlinear upgraded version of the constant strain triangular (CST) membrane finite element. Each load step is comprised of two iteration cycles: the geometrically nonlinear cycle that ensures equilibrium in the deformed sate using the Newton–Raphson iterations and the wrinkling cycle that identifies the location and direction of the wrinkles and redistributes stresses accordingly. Finally, validation and verification of the proposed analysis is made by comparing the present results with those existing for the benchmark examples.
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Badr, Elsayed, Sultan Almotairi, and Abdallah El Ghamry. "A Comparative Study among New Hybrid Root Finding Algorithms and Traditional Methods." Mathematics 9, no. 11 (June 7, 2021): 1306. http://dx.doi.org/10.3390/math9111306.
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In this paper, we propose a novel blended algorithm that has the advantages of the trisection method and the false position method. Numerical results indicate that the proposed algorithm outperforms the secant, the trisection, the Newton–Raphson, the bisection and the regula falsi methods, as well as the hybrid of the last two methods proposed by Sabharwal, with regard to the number of iterations and the average running time.
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Jalali-Vahid, D., H. Rahnejat, R. Gohar, and Z. M. Jin. "Prediction of oil-film thickness and shape in elliptical point contacts under combined rolling and sliding motion." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 214, no. 5 (May 1, 2000): 427–37. http://dx.doi.org/10.1243/1350650001543304.
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The paper presents a numerical solution for elliptical point contact conjunctions under combined rolling and sliding motion. This condition is prevalent in many practical applications, such as rolling element bearings and conformal gears. An effective influence Newton-Raphson method is employed in local point distributed or global line distributed low-relaxation iterations. This method enables determination of the pressure distribution and film shape at high loads such as are encountered in many practical applications. Some of the numerical predictions have been validated against experimental results.
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Esentürk, Emre, Nathan Luke Abraham, Scott Archer-Nicholls, Christina Mitsakou, Paul Griffiths, Alex Archibald, and John Pyle. "Quasi-Newton methods for atmospheric chemistry simulations: implementation in UKCA UM vn10.8." Geoscientific Model Development 11, no. 8 (August 1, 2018): 3089–108. http://dx.doi.org/10.5194/gmd-11-3089-2018.
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Abstract. A key and expensive part of coupled atmospheric chemistry–climate model simulations is the integration of gas-phase chemistry, which involves dozens of species and hundreds of reactions. These species and reactions form a highly coupled network of differential equations (DEs). There exist orders of magnitude variability in the lifetimes of the different species present in the atmosphere, and so solving these DEs to obtain robust numerical solutions poses a stiff problem. With newer models having more species and increased complexity, it is now becoming increasingly important to have chemistry solving schemes that reduce time but maintain accuracy. While a sound way to handle stiff systems is by using implicit DE solvers, the computational costs for such solvers are high due to internal iterative algorithms (e.g. Newton–Raphson methods). Here, we propose an approach for implicit DE solvers that improves their convergence speed and robustness with relatively small modification in the code. We achieve this by blending the existing Newton–Raphson (NR) method with quasi-Newton (QN) methods, whereby the QN routine is called only on selected iterations of the solver. We test our approach with numerical experiments on the UK Chemistry and Aerosol (UKCA) model, part of the UK Met Office Unified Model suite, run in both an idealised box-model environment and under realistic 3-D atmospheric conditions. The box-model tests reveal that the proposed method reduces the time spent in the solver routines significantly, with each QN call costing 27 % of a call to the full NR routine. A series of experiments over a range of chemical environments was conducted with the box model to find the optimal iteration steps to call the QN routine which result in the greatest reduction in the total number of NR iterations whilst minimising the chance of causing instabilities and maintaining solver accuracy. The 3-D simulations show that our moderate modification, by means of using a blended method for the chemistry solver, speeds up the chemistry routines by around 13 %, resulting in a net improvement in overall runtime of the full model by approximately 3 % with negligible loss in the accuracy. The blended QN method also improves the robustness of the solver, reducing the number of grid cells which fail to converge after 50 iterations by 40 %. The relative differences in chemical concentrations between the control run and that using the blended QN method are of order ∼ 10−7 for longer-lived species, such as ozone, and below the threshold for solver convergence (10−4) almost everywhere for shorter-lived species such as the hydroxyl radical.
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Novák, Josef P., Vlastimil Růžička, Anatol Malijevský, Jaroslav Matouš, and Jan Linek. "The DAN method for calculation of vapour-liquid equilibria using an equation of state; Bubble and dew point calculations." Collection of Czechoslovak Chemical Communications 50, no. 1 (1985): 1–22. http://dx.doi.org/10.1135/cccc19850001.
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A modification of the computational technique for calculating bubble and dew points using an equation of state has been proposed. The procedure consists in the Double Application of the Newton-Raphson method (DAN) to the set of equilibrium conditions. The algorithm is very effective as it provides both values of equilibrium variables and a very qualified first estimate of the next equilibrium point. This enables to proceed along the phase envelope rather quickly and to achieve convergence within a few iterations except in the close vicinity of the critical point.
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Hei, Di, Yong Fang Zhang, Mei Ru Zheng, Liang Jia, and Yan Jun Lu. "Stability and Bifurcation of Nonlinear Bearing-Flexible Rotor System with a Single Disk." Advanced Materials Research 148-149 (October 2010): 141–46. http://dx.doi.org/10.4028/www.scientific.net/amr.148-149.141.
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Dynamic model and equation of a nonlinear flexible rotor-bearing system are established based on rotor dynamics. A local iteration method consisting of improved Wilson-θ method, predictor-corrector mechanism and Newton-Raphson method is proposed to calculate nonlinear dynamic responses. By the proposed method, the iterations are only executed on nonlinear degrees of freedom. The proposed method has higher efficiency than Runge-Kutta method, so the proposed method improves calculation efficiency and saves computing cost greatly. Taking the system parameter ‘s’ of flexible rotor as the control parameter, nonlinear dynamic responses of rotor system are obtained by the proposed method. The stability and bifurcation type of periodic responses are determined by Floquet theory and a Poincaré map. The numerical results reveal periodic, quasi-periodic, period-5, jump solutions of rich and complex nonlinear behaviors of the system.
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Hassane Maina, Fadji, and Philippe Ackerer. "Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation." Hydrology and Earth System Sciences 21, no. 6 (June 8, 2017): 2667–83. http://dx.doi.org/10.5194/hess-21-2667-2017.
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Abstract. The solution of the mathematical model for flow in variably saturated porous media described by the Richards equation (RE) is subject to heavy numerical difficulties due to its highly nonlinear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the time-step magnitude within the iteration procedure while the nonlinear parameters are computed with the state variable at the previous time. The Ross method is an example of this type of scheme, and we show that it is equivalent to the Newton–Raphson method with a time-adaptive algorithm.Both algorithms are coupled to different time-stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change in water saturation. Three different test cases are used to evaluate the efficiency of these algorithms.The numerical results highlight the necessity of implementing an estimate of the time truncation errors.
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Brkić, Dejan, and Pavel Praks. "Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks." Applied Sciences 9, no. 10 (May 16, 2019): 2019. http://dx.doi.org/10.3390/app9102019.
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Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method that considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton–Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton–Raphson iterative method.
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Clarke, Brenton R. "Asymptotic theory for description of regions in which Newton-Raphson iterations converge to location M-estimators." Journal of Statistical Planning and Inference 15 (January 1986): 71–85. http://dx.doi.org/10.1016/0378-3758(86)90086-8.
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Sauve´, R. G., and D. R. Metzger. "Advances in Dynamic Relaxation Techniques for Nonlinear Finite Element Analysis." Journal of Pressure Vessel Technology 117, no. 2 (May 1, 1995): 170–76. http://dx.doi.org/10.1115/1.2842106.
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Traditionally, the finite element technique has been applied to static and steady-state problems using implicit methods. When nonlinearities exist, equilibrium iterations must be performed using Newton-Raphson or quasi-Newton techniques at each load level. In the presence of complex geometry, nonlinear material behavior, and large relative sliding of material interfaces, solutions using implicit methods often become intractable. A dynamic relaxation algorithm is developed for inclusion in finite element codes. The explicit nature of the method avoids large computer memory requirements and makes possible the solution of large-scale problems. The method described approaches the steady-state solution with no overshoot, a problem which has plagued researchers in the past. The method is included in a general nonlinear finite element code. A description of the method along with a number of new applications involving geometric and material nonlinearities are presented.
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Cai, Song Bai, Da Zhi Li, Chang Wan Kim, and Pu Sheng Shen. "A Simple Numerical Solution Procedure for Equations of Nonlinear Finite Element Method." Advanced Materials Research 889-890 (February 2014): 187–90. http://dx.doi.org/10.4028/www.scientific.net/amr.889-890.187.
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The conventional solution strategy for nonlinear FEM of structural analysis is usually based on Newton-Raphson iteration under an additional constraint equation. So far a lot of nonlinear finite element solution procedures have been devised to provide the basis for most nonlinear finite element computer programs. In order to produce effective, robust solution algorithms, additional constraint equations for nonlinear FEM calculations in the load-displacement space of has been extensively investigated for the last a few decades. However, it is widely believed that due to the additional computations in the controlling of steps and directions of the iteration procedure, there will be more round-off errors accumulated to influence the convergence of solution. In this work, a more simplified solution procedure is presented, which is featured to be with neither iterations nor constraints. A Fortran computer program of the algorithm presented has been implemented in combining with a space truss element of co-rotational procedure. Verification of the procedure has been done by numerical example and a good result achieved.
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Lu, Y. J., R. Dai, D. Hei, Y. F. Zhang, H. Liu, and L. Yu. "Stability and bifurcation of a non-linear bearing-flexible rotor coupling dynamic system." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 223, no. 4 (December 11, 2008): 835–49. http://dx.doi.org/10.1243/09544062jmes1190.
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The non-linear coupling dynamic behaviour of a hydrodynamic bearing-flexible rotor system is analysed. A local iteration method consisting of the improved Wilson-θ method, the predictor—corrector mechanism, and the Newton—Raphson method is proposed to calculate the non-linear dynamic response. Using the proposed method, the iterations are executed only on non-linear degrees of freedom. The iteration process shows improved convergence by taking the prediction value as the initial value. The stability and bifurcation type of periodic responses are determined by the Floquet theory. According to the physical characteristics of the oil film, a variational constraint approach is proposed to revise continuously the variational form of the Reynolds equation at each step of the iterative process. Non-linear oil film forces and their Jacobians are calculated simultaneously without an increase in computational costs, and compatible accuracy is obtained. Numerical results reveal periodic, quasi-periodic, coexisting, and jump solutions of rich and complex non-linear behaviours of the system and show that the proposed methods not only save computational costs but also have high precision.
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Moroz, Leonid V., Volodymyr V. Samotyy, and Oleh Y. Horyachyy. "Modified Fast Inverse Square Root and Square Root Approximation Algorithms: The Method of Switching Magic Constants." Computation 9, no. 2 (February 17, 2021): 21. http://dx.doi.org/10.3390/computation9020021.
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Many low-cost platforms that support floating-point arithmetic, such as microcontrollers and field-programmable gate arrays, do not include fast hardware or software methods for calculating the square root and/or reciprocal square root. Typically, such functions are implemented using direct lookup tables or polynomial approximations, with a subsequent application of the Newton–Raphson method. Other, more complex solutions include high-radix digit-recurrence and bipartite or multipartite table-based methods. In contrast, this article proposes a simple modification of the fast inverse square root method that has high accuracy and relatively low latency. Algorithms are given in C/C++ for single- and double-precision numbers in the IEEE 754 format for both square root and reciprocal square root functions. These are based on the switching of magic constants in the initial approximation, depending on the input interval of the normalized floating-point numbers, in order to minimize the maximum relative error on each subinterval after the first iteration—giving 13 correct bits of the result. Our experimental results show that the proposed algorithms provide a fairly good trade-off between accuracy and latency after two iterations for numbers of type float, and after three iterations for numbers of type double when using fused multiply–add instructions—giving almost complete accuracy.
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Danchick, Roy. "Gauss meets Newton again: How to make Gauss orbit determination from two position vectors more efficient and robust with Newton–Raphson iterations." Applied Mathematics and Computation 195, no. 2 (February 2008): 364–75. http://dx.doi.org/10.1016/j.amc.2007.03.053.
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Darmawan, Indra, Fuad Abdurrahman Rafif, and Addien Wahyu Wiratama. "SIMULASI ALIRAN DAYA PADA PEMBANGKIT LISTRIK TENAGA MESIN GAS (PLTMG) BADAS UNTUK SISTEM KELISTRIKAN SUMBAWA BESAR MENGGUNAKAN SOFTWARE ETAP 16." Jurnal Informatika, Teknologi dan Sains 1, no. 2 (November 29, 2019): 111–16. http://dx.doi.org/10.51401/jinteks.v1i2.416.
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Electric power system is a process of electrical energy generation to dispense to users. A well-integrated system provides quality assurance for system reliability. Analysis is conducted to minimize dispensing losses. The electrical system of Sumbawa Besar is currently in the process of upgrading 150 KV transmission distribution capacity with PLTMG BADAS as the largest generation center. The research was conducted by analyzing the power flow on PLTMG BADAS for the electrical system of Sumbawa Besar which was reviewed on 20 KV bus system, and by simulating using ETAP 16 software . The research method is done with data derived from Sumbawa's PLN and analysis of the power flow calculation using the method Newton-Raphson precision value 0.0001. The simulation is performed on three generation scenarios, namely: 1) Operation 1 PLTMG engine, 2) operation 2 PLTMG engines, 3) operation of the entire PLTMG machine. All three scenarios are done under normal operating conditions. Simulated results show that the analysis of power flows by Newton-Raphson method generates 2 iterations in each scenario, and the following parameters are obtained: 1) average of the highest voltage value in Scenario 1 OF 20.43 KV With phase angle-14.3, 2) Total active power – highest reactive in Scenario 2 of 44.475 MW and 9.557 MVAR, 3) Scenario 1 has the lowest power loss of 2.634 MW and 5.168 MVAR and voltage fall by 2.275%.
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Moosavian, Naser. "Pipe network modeling for analysis of flow in porous media." Canadian Journal of Civil Engineering 46, no. 12 (December 2019): 1151–59. http://dx.doi.org/10.1139/cjce-2018-0786.
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In this paper, a new matrix framework has been developed for the simulation of flow and pressure in porous media. In this framework, the pressure gradient formulation in Darcy’s law is considered as the head-loss equation in pipe network modeling. Then, an artificial pipe network has been constructed to find the pressure head profile in porous media. Two explicit and implicit formulations have been advanced for linear and nonlinear analysis, which the latter is an implementation of the Newton–Raphson algorithm. Both formulations iteratively solve a linear system of equations for calculating the nodal heads and apply a matrix multiplication for updating the flow vector. While the explicit method needs few iterations, the implicit method requires at least 20 iterations to converge with acceptable accuracy. For testing these formulations, four different types of network configurations were tested. The analysis of three laboratory tests showed that the application of the implicit method provides reliable and accurate results.
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Sujecki, Sławomir. "An Efficient Algorithm for Steady State Analysis of Fibre Lasers Operating under Cascade Pumping Scheme." International Journal of Electronics and Telecommunications 60, no. 2 (June 1, 2014): 143–49. http://dx.doi.org/10.2478/eletel-2014-0017.
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Abstract We derive an efficient algorithm for the steady state analysis of fibre lasers operating under cascade pumping scheme by combining the shooting method with the Newton-Raphson method. We compare the proposed algorithm with the two standard algorithms that have been used so far in the available literature: the relaxation method and the coupled solution method. The results obtained show that the proposed shooting method based algorithm achieves much faster convergence rate at the expense of a moderate increase in the calculation time. It is found that a further improvement in the computational efficiency can be achieved by using few iterations of the relaxation method to calculate the initial guess for the proposed shooting method based algorithm
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Jiang, L., and M. W. Chernuka. "A CO-ROTATIONAL FORMULATION FOR GEOMETRICALLY NONLINEAR FINITE ELEMENT ANALYSIS OF SPATIAL BEAMS." Transactions of the Canadian Society for Mechanical Engineering 18, no. 1 (March 1994): 65–88. http://dx.doi.org/10.1139/tcsme-1994-0005.
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A co-rotational procedure is presented in this paper for handling arbitrarily large three-dimensional rotations associated with geometrically nonlinear analysis of spatial beam structures. This procedure has been incorporated into two commonly used 3-D beam elements, the 2-node cubic beam element and the 3-node superparametric beam element, in our in-house general purpose finite element program, VAST. In the present procedure, the element tangent stiffness matrices are generated by using the standard updated Lagrangian formulation, while a co-rotational formulation is employed to update the internal force vectors during the Newton-Raphson iterations, A number of example problems have been analyzed and the result are in good agreement with analytical or published numerical solutions.
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Putri, Maulia, and Syaharuddin Syaharuddin. "Implementations of Open and Closed Method Numerically: A Non-linear Equations Solution Convergence Test." IJECA (International Journal of Education and Curriculum Application) 2, no. 2 (August 30, 2019): 1. http://dx.doi.org/10.31764/ijeca.v2i2.2041.
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Non-linear equations are one of the studies in mathematics. Root search in complex non-linear equations can be solved by numerical methods. Many methods to solve the equation. Therefore, the purpose of this research is to conduct simulation of closed and open methods such as Newton Raphson method, Secant method, Regula Falsi, Fixet Point, and Bisection. This is done as a form of comparative research to see the accuracy, number of iterations, and errors of each method in resolving the non-linear equations. As for the case being resolved is the roots of the exponential equation, trigonometry, logarithmic and polynomial degrees of three. The results of this study resulted in different levels of convergence in resolving each case
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Praks, Pavel, and Dejan Brkić. "Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation." Processes 6, no. 8 (August 16, 2018): 130. http://dx.doi.org/10.3390/pr6080130.
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The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.
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Sabharwal. "Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms." Mathematics 7, no. 11 (November 16, 2019): 1118. http://dx.doi.org/10.3390/math7111118.
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Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.
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Wu, Peng, Shaojing Su, Zhen Zuo, Xiaojun Guo, Bei Sun, and Xudong Wen. "Time Difference of Arrival (TDoA) Localization Combining Weighted Least Squares and Firefly Algorithm." Sensors 19, no. 11 (June 4, 2019): 2554. http://dx.doi.org/10.3390/s19112554.
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Time difference of arrival (TDoA) based on a group of sensor nodes with known locations has been widely used to locate targets. Two-step weighted least squares (TSWLS), constrained weighted least squares (CWLS), and Newton–Raphson (NR) iteration are commonly used passive location methods, among which the initial position is needed and the complexity is high. This paper proposes a hybrid firefly algorithm (hybrid-FA) method, combining the weighted least squares (WLS) algorithm and FA, which can reduce computation as well as achieve high accuracy. The WLS algorithm is performed first, the result of which is used to restrict the search region for the FA method. Simulations showed that the hybrid-FA method required far fewer iterations than the FA method alone to achieve the same accuracy. Additionally, two experiments were conducted to compare the results of hybrid-FA with other methods. The findings indicated that the root-mean-square error (RMSE) and mean distance error of the hybrid-FA method were lower than that of the NR, TSWLS, and genetic algorithm (GA). On the whole, the hybrid-FA outperformed the NR, TSWLS, and GA for TDoA measurement.
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Tommasini, D., and D. N. Olivieri. "Comment on ‘An efficient code to solve the Kepler equation: elliptic case’." Monthly Notices of the Royal Astronomical Society 506, no. 2 (June 24, 2021): 1889–95. http://dx.doi.org/10.1093/mnras/stab1790.
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ABSTRACT In a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a scheme for obtaining very close seeds for solving the elliptic Kepler equation with the classical and modified Newton–Raphson methods. This implied an important reduction in the number of iterations needed to reach a given accuracy. However, RPP also made strong claims about the errors of their method that are incorrect. In particular, they claim that their accuracy can always reach the level of ∼5ε, where ε is the machine epsilon (e.g. ε = 2.2 × 10−16 in double precision), and that this result is attained for all values of the eccentricity e < 1 and the mean anomaly M ∈ [0, π], including for e and M that are arbitrarily close to 1 and 0, respectively. However, we demonstrate both numerically and analytically that any implementation of the classical or modified Newton–Raphson methods for Kepler’s equation, including those described by RPP, has a limiting accuracy of the order of ${\sim}\varepsilon /\sqrt{2(1-e)}$. Therefore the errors of these implementations diverge in the limit e → 1, and differ dramatically from the incorrect results given by RPP. Despite these shortcomings, the RPP method can provide a very efficient option for reaching such limiting accuracy. We also provide a limit that is valid for the accuracy of any algorithm for solving Kepler equation, including schemes like bisection that do not use derivatives. Moreover, similar results are also demonstrated for the hyperbolic Kepler equation. The methods described in this work can provide guidelines for designing more accurate solutions of the elliptic and hyperbolic Kepler equations.
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Korczyński, Adam. "Predicting in multivariate incomplete time series. Application of the expectation-maximisation algorithm supplemented by the Newton-Raphson method." Przegląd Statystyczny 68, no. 1 (August 24, 2021): 17–46. http://dx.doi.org/10.5604/01.3001.0015.0376.
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Statistical practice requires various imperfections resulting from the nature of data to be addressed. Data containing different types of measurement errors and irregularities, such as missing observations, have to be modelled. The study presented in the paper concerns the application of the expectation-maximisation (EM) algorithm to calculate maximum likelihood estimates, using an autoregressive model as an example. The model allows describing a process observed only through measurements with certain level of precision and through more than one data series. The studied series are affected by a measurement error and interrupted in some time periods, which causes the information for parameters estimation and later for prediction to be less precise. The presented technique aims to compensate for missing data in time series. The missing data appear in the form of breaks in the source of the signal. The adjustment has been performed by the EM algorithm to a hybrid version, supplemented by the Newton-Raphson method. This technique allows the estimation of more complex models. The formulation of the substantive model of an autoregressive process affected by noise is outlined, as well as the adjustment introduced to overcome the issue of missing data. The extended version of the algorithm has been verified using sampled data from a model serving as an example for the examined process. The verification demonstrated that the joint EM and Newton-Raphson algorithms converged with a relatively small number of iterations and resulted in the restoration of the information lost due to missing data, providing more accurate predictions than the original algorithm. The study also features an example of the application of the supplemented algorithm to some empirical data (in the calculation of a forecasted demand for newspapers).
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Zhao, Haining, Hongbin Jing, Zhengbao Fang, and Hongwei Yu. "Flash Calculation Using Successive Substitution Accelerated by the General Dominant Eigenvalue Method in Reduced-Variable Space: Comparison and New Insights." SPE Journal 25, no. 06 (July 20, 2020): 3332–48. http://dx.doi.org/10.2118/202472-pa.
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Summary On the basis of a previously published reduced-variables method, we demonstrate that using these reduced variables can substantially accelerate the conventional successive-substitution iterations in solving two-phase flash (TPF) problems. By applying the general dominant eigenvalue method (GDEM) to the successive-substitution iterations in terms of the reduced variables, we obtained a highly efficient solution for the TPF problem. We refer to this solution as Reduced-GDEM. The Reduced-GDEM algorithm is then extensively compared with more than 10 linear-acceleration and Newton-Raphson (NR)-type algorithms. The initial equilibrium ratio for flash calculation is generated from reliable phase-stability analysis (PSA). We propose a series of indicators to interpret the PSA results. Two new insights were obtained from the speed comparison among various algorithms and the PSA. First, the speed and robustness of the Reduced-GDEM algorithm are of the same level as that of the reduced-variables NR flash algorithm, which has previously been proved to be the fastest flash algorithm. Second, two-side phase-stability-analysis results indicate that the conventional successive-substitution phase-stability algorithm is time consuming (but robust) at pressures and temperatures near the stability-test limit locus in the single-phase region and near the spinodal in the two-phase region.
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Lu, Xiao Yang, Xiao Li Lu, Bing Tao Tang, and Li Li Huang. "Novel Plasticity Integration Algorithm in Improved Inverse Analysis Method and its Verification in Clover-Shaped Cup Drawing." Advanced Materials Research 366 (October 2011): 121–26. http://dx.doi.org/10.4028/www.scientific.net/amr.366.121.
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An improved inverse analysis method is developed based on the final workpiece in Euler coordinate system. The drawbeads and the radius of the die introduce a complex bending-unbending loading history as the material passes through these regions. Unlike the widespread inverse analysis using deformation theory of plasticity, in order to consider loading history, the improved inverse analysis method uses the constitutive equation based on flow theory of plasticity. In order to avoid numerous iterations to ensure the numerical stability in Newton-Raphson scheme to obtain plastic multiplier , a novel plastic integration algorithm is proposed to consider bending–unbending effects. A clover-shaped cup drawing example is numerically simulated with the inverse analysis method based on deformation theory of plasticity and the improved one based on flow theory of plasticity. These simulated results are compared with those of the incremental forward finite element solver LS-DYNA simultaneously. The comparisons of blank configurations and the effective strain distribution show that the proposed plasticity integration algorithm is effective and reliable.
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Meramo-Hurtado, Samir, Plinio Puello, and Julio Rodriguez. "Application of Solution Strategies for Numerical Estimation of Thermodynamic Equilibrium Parameters for an Acetone–Butanol Mixture." Applied Sciences 10, no. 9 (April 30, 2020): 3136. http://dx.doi.org/10.3390/app10093136.
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The development of reliable numerical estimation of thermodynamic parameters is a crucial aspect in the ongoing research about process engineering and design. The consideration of these concepts lets to design more precise processing units and separations stages based on the predicted nature of substances. Therefore, this study presents an application of different solution methods for the estimation of thermodynamic equilibrium parameters of an acetone–butanol mixture. This dissolution is a non-ideal system, so, the non-ideal Raoult’s Law and Wilson’s equation were used to model the liquid–vapor equilibrium. Otherwise, the solution of this system required the application of nonlinear least squares (NLS) for determination of adjustable parameters. As the above step transformed Wilson’s equation into a system of nonlinear equations, solution algorithms such as; Newton–Raphson method (NRM), Broyden’s method (BM) and Levenberg–Marquardt method (LMM) were applied. All algorithms converged towards the same solution ( Λ 12 = 0.689 and Λ 21 = 0.798 ), but Newton’s and Broyden’s methods employed fewer computational time and number of iterations compared to performance showed by the Levenberg–Marquardt algorithm.
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Shabana, A. "Dynamics of Inertia-Variant Flexible Systems Using Experimentally Identified Parameters." Journal of Mechanisms, Transmissions, and Automation in Design 108, no. 3 (September 1, 1986): 358–66. http://dx.doi.org/10.1115/1.3258740.
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In this investigation modal parameters (frequency, damping, and mode shapes) which are determined experimentally using parameter estimation techniques are employed to simulate and predict the dynamic behavior of flexible multibody systems which consist of interconnected rigid and flexible components. The system differential equations of motion and algebraic constraint equations describing mechanical joints in the system are first identified using analytical techniques. Dynamic parameters such as mass, damping, and stiffness coefficients that appear in the system differential equations are then identified using a set of experimentally measured data. Mode shapes which are the result of the experimental identification are used to write the physical elastic coordinates of selected nodal points on the flexible body in terms of a reduced set of modal coordinates. The nonlinear differential and algebraic constraint equations are then written in terms of mixed sets of coupled reference and modal coordinates. These equations are integrated numerically using a direct numerical integration technique coupled with Newton–Raphson type iterations in order to check on constraint violations. The formulation developed is numerically exemplified using a three-dimensional dune buggy vehicle model.
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Souza, Luiz Antonio Farani de, Leandro Vanalli, and Arthur Bueno de Luz. "Numerical-Computational Model for Nonlinear Analysis of Frames with Semirigid Connection." Mathematical Problems in Engineering 2020 (September 3, 2020): 1–11. http://dx.doi.org/10.1155/2020/3613892.
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A numerical-computational model for static analysis of plane frames with semirigid connections and geometric nonlinear behavior is presented. The set of nonlinear equations governing the structural system is solved by the Potra–Pták method in an incremental procedure, with order of cubic convergence, combined with the linear arc-length path-following technique. The algorithm pseudo-code is presented, and the finite element corotational method is used for the discretization of the structures. The equilibrium paths with load and displacement limit points are obtained. The semirigidity is simulated by a linear connection element of null length, which considers the axial, tangential, and rotational stiffness. Nonlinear analyses of 2D frame structures are carried out with the free Scilab program. The results show that the Potra–Pták procedure can decrease the number of iterations and the computing time in comparison with the standard and modified Newton–Raphson iterative schemes. Also, the simulations show that the connection flexibility has a strong influence on the nonlinear behavior and stability of the structural systems.
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Praks, Pavel, and Dejan Brkić. "Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction." Advances in Civil Engineering 2018 (December 10, 2018): 1–18. http://dx.doi.org/10.1155/2018/5451034.
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The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.
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Munhoven, Guy. "SolveSAPHE-r2 (v2.0.1): revisiting and extending the Solver Suite for Alkalinity-PH Equations for usage with CO<sub>2</sub>, HCO<sub>3</sub><sup>−</sup> or CO<sub>3</sub><sup>2−</sup> input data." Geoscientific Model Development 14, no. 7 (July 6, 2021): 4225–40. http://dx.doi.org/10.5194/gmd-14-4225-2021.
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Abstract. The successful and efficient approach at the basis of the Solver Suite for Alkalinity-PH Equations (SolveSAPHE) (Munhoven, 2013), which determines the carbonate system speciation by calculating pH from total alkalinity (AlkT) and dissolved inorganic carbon (CT), and which converges for any physically sensible pair of such data, has been adapted and further developed to work with AlkT–CO2, AlkT–HCO3-, and AlkT–CO32-. The mathematical properties of the three modified alkalinity–pH equations are explored. It is shown that the AlkT–CO2, and AlkT–HCO3- problems have one and only one positive root for any physically sensible pair of data (i.e. such that [CO2]>0 and [HCO3-]>0). The space of AlkT–CO32- pairs is partitioned into regions where there is either no solution, one solution or where there are two. The numerical solution of the modified alkalinity–pH equations is far more demanding than that for the original AlkT–CT pair as they exhibit strong gradients and are not always monotonous. The two main algorithms used in SolveSAPHE v1 have been revised in depth to reliably process the three additional data input pairs. The AlkT–CO2 pair is numerically the most challenging. With the Newton–Raphson-based solver, it takes about 5 times as long to solve as the companion AlkT–CT pair; the AlkT–CO32- pair requires on average about 4 times as much time as the AlkT–CT pair. All in all, the secant-based solver offers the best performance. It outperforms the Newton–Raphson-based one by up to a factor of 4 in terms of average numbers of iterations and execution time and yet reaches equation residuals that are up to 7 orders of magnitude lower. Just like the pH solvers from the v1 series, SolveSAPHE-r2 includes automatic root bracketing and efficient initialisation schemes for the iterative solvers. For AlkT–CO32- data pairs, it also determines the number of roots and calculates non-overlapping bracketing intervals. An open-source reference implementation of the new algorithms in Fortran 90 is made publicly available for usage under the GNU Lesser General Public Licence version 3 (LGPLv3) or later.